Well-posedness analysis and pseudo-Galerkin approximations using Tau Legendre algorithm for fractional systems of delay differential models regarding Hilfer (α,β)-framework set

Fractional calculus serves as a versatile and potent tool for the modeling and control of intricate systems. This discussion debates the system of DFDEs with two regimes; theoretically and numerically. For theoretical analysis, we have established the EUE by leveraging the definition of Hilfer (α,β)-framework. Our investigation involved the examination of the possessions of the FRD, FCD, and FHD, utilizing their forcefulness and qualifications to convert the concerning delay system into an equivalent one of fractional DVIEs. By employing the CMT, we have successfully demonstrated the prescribed requirements. For numerical analysis, the Galerkin algorithm was implemented by leveraging OSLPs as a base function. This algorithm allows us to estimate the solution to the concerning system by transforming it into a series of algebraic equations. By employing the software MATHEMATICA 11, we have effortlessly demonstrated the requirements estimation of the nodal values. One of the key advantages of the deployed algorithm is its ability to achieve accurate results with fewer iterations compared to alternative methods. To validate the effectiveness and precision of our analysis, we conducted a comprehensive evaluation through various linear and nonlinear numerical applications. The results of these tests, accompanied by figures and tables, further support the superiority of our algorithm. Finally, an analysis of the numerical algorithm employed was provided along with insightful suggestions for potential future research directions.

century.This can be partly attributed to advancements in computing technology during this period, as they facilitated efficient solutions to fractional models [1][2][3].The applications of fractional patterns span various domains including engineering, signal processing, statistics, and others.For instance, fractional calculus finds utility in modeling the behavior of viscoelastic materials, controlling robotic systems, predicting financial markets, and modeling biological systems [4][5][6][7].
The DFDEs are a class of ordinary calculus that emphasizes fractional derivatives and delays.The scope of DFDEs is more comprehensive than traditional delay models, allowing for a broader range of phenomena to be represented.Applications of DFDEs appear in diverse domains involving signal processing, statistics, engineering, and biology where various simulation techniques are used [8][9][10][11][12][13][14][15][16][17][18][19].For instance, in physics, DFDEs have been used to model heat transfer, diffusion, and wave propagation in materials with memory properties [20,21].In engineering, DFDEs have proven useful for modeling and controlling systems such as robots, aircraft, and power networks [7,[22][23][24].Within biology, DFDEs have been employed to formulate agent-based representations of systems like the cardiovascular, nervous, and immune systems [25,26].
In this particular research, we adopted the parameterization of OSLPs to substitute the necessary background functions in (1).Subsequently, the SGLA was employed to convert (1) into sets of algebraic equations.Upon handling the producing set, the desired approximate solution was obtained.One notable numerical optimality of the SGLA employed is its feasibility for any model formalism.Furthermore, it enumerates significantly valid approximations by leveraging a minimal number of OSLP terms.
The organization of the computations and algorithm development are ordered as next.Section 2 presents spectrum attributes, and necessary lemmas based on the definitions of FRD, FCD, and FHD.In Section 3, we transform the system described in (1) and ( 2) subjected to a congruent set of DVIEs.Additionally, this section outlines the development of proof for the EUE concerning (1) and (2).In Section 4, we introduce the SGLA as a solution technique for tackling (1) and ( 2) and proceed to characterize the theorems on convergence.Section 5 provides numerical applications and results.Finally, a summary of key findings is mentioned in Section 6.

Background and overview results
This section presents indispensable background and properties associated with FRD, FCD, and FHD approaches, simultaneously, with significant outcomes that will be exploited in the next portion.Anyhow, consider the following requirements: • The Banach topological manifold L P (J,R) is axiomatized as the collection of each Lebesgue measurable S : • The topological manifold ACðJ; RÞ ¼ fS : J ! R; S absolutely continuous on Jg.Indeed, the topological manifold AC ℏ ðJ; RÞ ¼ fS : S 2 C ℏÀ 1 ðJ; RÞ and S ℏÀ 1 2 ACðJ; RÞg.

Congruence fractional DVIE and the results concerning EUE
In this section, we investigate the examination of the possessions of the FRD, FCD, and FHD utilizing their forcefulness and qualifications to convert the concerning delay system into an equivalent one of fractional DVIEs.After that, we visualize and clarify the EUE concerning (1) and (2) leveraging its congruence system of fractional DVIEs (9) within C g 1À g ðJÞ.

The SGLA: Assembly and outcomes
This section introduces the SGLA numerical scheme for solving (1) and ( 2).This approach is a type of weighted residual numerical technique that uses a finite set of basis polynomials as weighting functions.Based on our proposed algorithm, we employ orthogonal spline local polynomial functions as the weighting functions.Indeed, we prove theorems regarding the convergence and error estimation of the SGLA.Broadly, the SGLA is a numerical technique used to solve various classes of differential/integral problems including several types by approximating the solution by leveraging a finite collection of orthogonal basis functions.In implementing the Galerkin scheme, the orthogonal basis functions are selected to satisfy the initial or boundary constraints.The effectiveness of the Galerkin approach relies on several prerequisites as follows: • The basis functions must be orthogonal, facilitating straightforward computation of solution coefficients.
• The selected basis functions should have regular behavior and adequate smoothness to ensure precise approximation.
• The Galerkin numerical scheme will provide accurate results if the problem has a! -Sol that depends continuously on the data and problem constraints.Failure to meet these constraints may result in imprecise or erroneous outcomes from the Galerkin approach.
Initially, we define the OSLPs as basis functions for representing functions within a predefined interval.Following that, we use this concept to define the essential functions in (1) in terms of OSLPs.Next, we calculate the residual required in the Galerkin numerical process.Leveraging the orthogonality of the residual with the OSLPs, the sets of FDDSs can be simplified into an algebraic one.Finally, this system can be solved by leveraging MATHEMATICA 11 to obtain the needed approximation.Definition 5. [34] The OSLP of multiplicity ϐ is The condition of orthogonality is defined as

Computational demonstrations
Herein, we solve two systems of DFDEs in the FHD sense that involve various instances by leveraging the proposed algorithm.While the second example is nonlinear, the first one is linear.Our methodology involves controlling the relative error R ℏ ðƻÞ ¼ ðR ℏ;1 ðƻÞ; R ℏ;2 ðƻÞÞ and the absolute error A ℏ ðƻÞ ¼ ðA ℏ;1 ðƻÞ; A ℏ;2 ðƻÞÞ for various values of α, β, and ℏ.We provide a thorough examination of the effectiveness through plots and data that contrast the exact solution with the approximations.

SGLA: Steps and examples
The SGLA provides several benefits and utilities when handling a set of FDEs as • The simulated outcomes achieved closely approximates the exact solution.As evidenced by the obtained R ℏ (ƻ) and A ℏ ðƻÞ, SGLA delivers highly sufficient and accurate results.
• The SGLA can achieve high validity by leveraging tiny iterations in the OSLP expansions.
• The SGLA represents a straightforward scheme to incorporate that does not require sophisticated mathematical apparatuses or a proficient programmer.
• The utilized SGLA serves as a universal technique that can be implemented to exhibit other instances of fractional systems.
• The core characteristic of SGLA is its applicability to other orthogonal basis functions.
Algorithm 1 clearly outlines the steps to derive a solution leveraging the presented SGLA and the specified FHD.At this point, either a programmer or a specialized mathematician with relevant expertise could implement this algorithm into a program by leveraging the MATHEMATICA 11 environment, consistent with how we demonstrated this approach in this paper.Let us begin our calculations with the following two suitable instances, both of which have precise solutions throughout the chosen integral domain, ensuring the accuracy of the data outputs.

Analysis of the findings
This study applies the preceding numerical technique to approximate solutions to (1) and ( 2).Graphical and tabulated outputs are generated for each example scenario under varied configurations of {α,β} and ℏ parameters.Additionally, computations of A ℏ ðƻÞ and R ℏ (ƻ) are carried out and compared extensively, to evaluate the efficacy of the SGLA numerical approach.
For Example 2; in Table 3, the correlation between data operations A ℏ;2 ðƻÞ and R ℏ,1 (ƻ) are presented at ℏ = 3, β = 0.5, and α2{0.1,0.7}.In Table 4, the correlation between data operations From the charts in the antecedent figures, we interestingly that the apparent drawing curves are very low towards the ƻ-axis.This confirms the data of the driving tables.In addition, these curves contain oscillations at their beginning or end, which confirms the occurrence of a delay in the fraction solutions, and also confirms the shape of the previously selected examples.

Summary of key findings
Throughout the work, we established the EUE concerning a set of DFDEs in the sense of the FHD approach.Our investigation involved examining the properties of the FRD, FCD, and FHD, utilizing their hallmarks to convert our system of DFDEs into an equivalent one of fractional DVIE with the help of the CMT.For numerical analysis, we implemented the SGLA leveraging OSLPs.This algorithm allows us to approximate the solution to the considered system by transforming it into a series of algebraic equations.One of the key advantages of this algorithm is its ability to achieve accurate results with fewer iterations compared to alternative   methods.The numerical convergence and error estimates of the approach are examined and to validate the effectiveness and accuracy of our algorithm, we conducted a comprehensive evaluation through various linear and nonlinear numerical applications.The data from these evaluations, accompanied by charts and tables, further validate the strength of our algorithmic approach.Future research directions for our simulations could include applying it to additional types of DFDEs, such as those containing multiple delays or nonlinear terms.The algorithm could also be analyzed for solving DFDEs and fractional delay integral models.Furthermore, this research emphasizes the adaptability of the SGLA, as it can be extended to other complex fractional problems and systems by utilizing alternative orthogonal basis functions like Chebyshev or Fourier polynomials.Additionally, the algorithm may be parallelized to improve computational efficiency.